CD8+ T cells control SIV infection using both cytolytic effects and non-cytolytic suppression of virus production

Whether CD8+ T lymphocytes control human immunodeficiency virus infection by cytopathic or non-cytopathic mechanisms is not fully understood. Multiple studies highlighted non-cytopathic effects, but one hypothesis is that cytopathic effects of CD8+ T cells occur before viral production. Here, to examine the role of CD8+ T cells prior to virus production, we treated SIVmac251-infected macaques with an integrase inhibitor combined with a CD8-depleting antibody, or with either reagent alone. We analyzed the ensuing viral dynamics using a mathematical model that included infected cells pre- and post- viral DNA integration to compare different immune effector mechanisms. Macaques receiving the integrase inhibitor alone experienced greater viral load decays, reaching lower nadirs on treatment, than those treated also with the CD8-depleting antibody. Models including CD8+ cell-mediated reduction of viral production (non-cytolytic) were found to best explain the viral profiles across all macaques, in addition an effect in killing infected cells pre-integration (cytolytic) was supported in some of the best models. Our results suggest that CD8+ T cells have both a cytolytic effect on infected cells before viral integration, and a direct, non-cytolytic effect by suppressing viral production.


SUPPLEMENTARY NOTE 1 Model with Viral Dynamics Only
To further help interpret the viral load profiles during CD8 + cell depletion and RAL monotherapy, we also used the original slow and rapid integration virus dynamic model 1 without CD4 + T cell proliferation.In this case, we only fit the model to viral load observation and not to the Ki67 + CD4 + T cell count.Without this assumption, the model in equation (1) in the main manuscript has the form: To fit the model in equation S1 to the data, we used nonlinear mixed-effect modeling as described in the main text.Unlike in the main manuscript, here we only modeled the plasma viral load for animal  at time  as   = log 10 (  ) +   with   ~(0,   2 ) the error for the logged viral load.
We performed the data fitting in two steps, as this allowed better convergence of the parameter estimates.We first fitted the model in equation (S1) to the viral load data from the 8 animals under RAL monotherapy only.With this fit, we estimate parameter distributions for  1 ,  2 ,   to be used in the second step, we also fixed =10 -8 , as this parameter trades-off with TI in this simple model (see 1 for model details).Other parameters were fixed as in the main text.In these initial fits,  = 0 refers to initiation of RAL treatment.We assumed that the system in (S1) is in steady state before  = 0, allowing us to obtain the values of ( +  1 ),     (0) =   +  (0)  1 (0) and  = ( + (0))  (0).
In a second step, we performed fits of the model in equation (S1) to the viral load data from all 20 animals simultaneously, fixing the same parameters as before and fixing the distribution of  1 ,  2 ,   , as estimated in the RAL-only fits.We repeat each fit by assuming that CD8 + cell depletion has one or more of the following effects: (i) reduction of the death rate of short-lived infected cells before viral integration ( 1 ), (ii) reduction of the death rate of productively infected cells ( 2 ), (iii) increasing the viral infectivity rate (), or (iv) increasing the virus production rate ().These effects were simulated by changing the corresponding parameters in equation ( 1) under depletion conditions to (1 −  1 ) 1 , (1 −  2 ) 2 , (1 +  3 ) and (1 +  4 ) then estimating   , one at a time or in combination.We thus have 16 different models (Table S3), from all   = 0 (our original model) to all   ≠ 0 (indicating the CD8 + depletion affects each of the four parameters).Depending on the effect or combination of effects in each fit, we estimate the respective reduction or increase in each parameter (  ).We fit each CD8-depletion effect model to the whole data set 10 times allowing for random initial guesses of the parameters to be estimated.Each of these times, we estimated the log-likelihood (log ℒ).For the case with highest likelihood (max (log ℒ)) from the 10 fits, we then computed the corrected Akaike Information Criteria (AIC) as  = −2 max(log ℒ) + 2 + 2(+1) −−1 , where  is the number of parameters estimated and  de number of data points from all animals 2 .We used AIC to compare models with different CD8 + cell effects.

SUPPLEMENTARY NOTE 2 Model Identifiability
To assess the identifiability of our models, we used both an analytical approach 3 , which analyzes structural identifiability, and the profile likelihood method, [4][5][6] which provides information on both structural and practical identifiability.For the former, the idea is to find whether scaling of the parameters to be fitted and unobserved variables leaves the system invariant, exploring symmetries of the equations that are related to more sophisticated methods based on the theory of Lie groups. 3,4,7 the profile likelihood method, we used Monolix 8 with each parameter, in turn, fixed at a given value (iterated over a set of values) and fitting the other parameters to assess the loglikelihood of these fits with one less parameter (the fixed one).The 95% confidence interval for the fixed parameter, based on the chi-squared distribution with one degree of freedom is given by the values of the parameter where the -2 log-likelihood is 3.84 units larger than the minimum. 4

Model with Viral Dynamics Only
We first fitted the model to the data from the group of animals receiving RAL monotherapy only, which has also been done for HIV 9 .Virus kinetics analyses have shown that the different phases of viral decline after initiation of treatment can reveal the kinetics of infected cells 1,9 .Therefore, from these fits we estimated the death rate of infected cells before ( 1 ) and after integration ( 2 ).These estimates will be used as reference when analyzing the effects of CD8 + cell depletion.
We then fitted the model to the three treatment groups together to see what effect of CD8 + cell depletion best explained all the data.Figure S5 show the model predictions using the estimates of best fits to each group of animals.From the best fit, our model predicts that CD8 + cell depletion affects both the loss rate of infected cells before integration (effect in  1 ) and the rate of virus production (effect in ), with negligible effect in viral infectivity () and death rate of productively infected cells ( 2 ).

Model Identifiability
To test the structural identifiability of our model, we used a scaling method based on the invariance of the equations under transformation of the parameters 3 .In this method, one chooses scaling factors ui for each parameter and unobserved variable and then equate each term of the equations with and without the scaling factor.Those parameters for which the only scaling factor that satisfies these equations is ui=1 are structurally identifiable (see 3 for details).
As an example, let's analyze the first equation of our model, including the scaling factors.
Note that we only use the scaling factors for parameters to fit (so,  and d, which are kept constant are not multiplied by these factors) and to non-observed variables (so, TI+I1 is not multiplied because it is our Ki67 observed quantity).Now, dividing both sides by uT, the resulting right-hand side should be equal to the first equation of our model (in the main text) for dTI/dt.
Because the equations must be equal for all values of V(t), TI(t) and TI(t)+I1(t), we can equate each functionally independent term of the equations.Doing this we obtain , where the last result requires some simple algebra.Whenever the equalities are satisfied only for scaling factor, ui, equal to 1, the corresponding parameter is identifiable.For example, if the result of the equality was uv1uv2=1, then variables v1 and v2 would not be identifiable.
We proceeded in the same way systematically analyzing the other equations of our model Eq. ( 1) of the main text, for all fitted parameters and unobserved variables, and found that in all cases the scaling factor ui was 1.Thus, we conclude that our model is structurally identifiable without the factors for the effects of CD8 + cell depletion.When we add these factors, for example (1 -1) 1, we see that solution for the scaled equations can have terms of the form u1(1-u11)=1-1, which don't have a unique solution u1=1 and u1=1.We note that using mixed effects models adds some subtleties to the analysis of structural identifiability 3 , since we are fitting all the subjects at the same time, in this case with two slightly different models, with and without the i factors.Thus, for example, since 1 is identifiable from fitting the RAL only macaques, such that u1=1, then also the u1=1 and we recover identifiability.In fact, this is one of the big advantages of population fitting using mixed-effects models.
Next, we analyzed identifiability using the profile likelihood method, which addresses both structural and practical identifiability [4][5][6] .The results are presented in figure S7 (for the full model from the main text) and S8 (for the viral load only model from this supplementary text).
We found that all parameters for the best fitting model (in Table 1 of the main text) are identifiable (Figure S7), although for some parameters the fitting is not very stable with wide confidence intervals (e.g., parameter K).We also tested parameter 1 for the reduction in the death rate of pre-integration infected cells, which came out as a relevant parameter in the top models explaining the full data set (Supplementary Table 1).Although the best fits of the model corresponded to values of 1 very close to 1, this parameter is difficult to estimate with a flat profile likelihood, in a way consistent with the structural identifiability analysis mentioned above.
For the simpler model, which we fitted in two stages as described, all the fitted parameters were identifiable including 1, although the profile likelihoods still show some instability.

Table 2 . Individual parameter estimates for the model in which CD8 + cells affect death rate of infected cells prior to integration, CD4 + T cell proliferation and virus production.
For all animals the population estimates for the CD8+ cell effects were:  4 = 0.84 Note: The AICc value for the best model is 184.2.